The Syrian regime’s relentless assault on the rebel-held area of Eastern Ghouta outside Damascus has spared little, targeting not only hospitals and markets, but also medical rescuers and their families.

Residents and medical workers describe the offensive, backed by Russian airstrikes, as a war of extermination meant to bring the suburb to its knees and force its unconditional surrender. According to the White Helmets rescue group, the sustained attack has already killed more than 250 people since it began Monday morning,...